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<title>Atlas software user guide -- D_2n</title>
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<h2>Even rank type D</h2>
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<i>Last updated: October 8, 2005</i>
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Even rank type D is the type that causes by far the most headaches as far as
the classification of inner classes and real forms is concerned. 
<a href="D4.html">Type D4</a> is even a little bit more exceptional than
the others; therefore we assume in this page that we are in type D_2m, with
m > 2.
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As for inner classes, this is the only irreducible Dynkin diagram possessing
a non-trivial involution, for which the split form <b>so</b>(2m,2m) and the 
compact form <b>so</b>(4m) belong to the same inner class. The inner class
corresponding to this non-trivial involution must be accessed through the
&#8220;u&#8221; (unequal rank) symbol. The fundamental real form in this class 
is <b>so</b>(4m-1,1), and the quasisplit form is <b>so</b>(2m+1,2m-1).
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These groups are also the only ones for which the simply connected group
(i.e., <b>Spin</b>(4m)), has a center that is not cyclic, viz. 
<b>Z</b>/2.<b>Z</b>/2. This causes the printout of the center of the
simply connected group to have more cyclic factors than there are factors
in the Lie type, with the corresponding danger of confusion.
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As for real forms, these groups are also the only simple Lie types for which
there are distinct weak real forms corresponding to isomorphic real groups
(in other words, there are involutions of G that are conjugate in the outer
automorphism group but not in the inner one.) In fact, there is just one pair
of weak real forms that exhibits this behaviour, viz. the two 
&#8220;versions&#8221; of
<b>so</b>*(4m). In the program, we distinguish these as follows. There are
three non-trivial elements in the center, which we denote [0,1], [1,0] and
[1,1]. We arrange the correspondence in such a way that [1,1] is the element
that is fixed by the non-trivial element of Out(G) (recall that we have assumed
that we are not in <a href="D4.html">type D4</a>.) Then the quotient of
<b>Spin</b>(4m) by the two-element subgroup generated by [1,1] is 
<b>SO</b>(4m), and the real forms corresponding to the two <b>so</b>*'s
for this quotient are both connected. Now for each of the two other two-element
subgroups in the center, one of the <b>so</b>*'s gives rise to a non-connected
real form, and the other to a connected one. We denote <b>so</b>*(4m)[0,1] the
one that gives rise to a non-connected real form in the quotient by the
subgroup generated by [0,1], and <b>so</b>*(4m)[1,0] the other. (The 
computation of <a href="components.html">component groups</a> is one of the 
capabilities of the program.)
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